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A discrete-time model of American put option in an uncertain environment. (English) Zbl 1112.91328
Summary: A discrete-time mathematical model for American put option with uncertainty is presented, and the randomness and fuzziness are evaluated by both probabilistic expectation and fuzzy expectation defined by a possibility measure from the viewpoint of fuzzy expectation, taking account of decision-maker’s subjective judgment. An optimality equation for the optimal stopping problem in a fuzzy stochastic process is derived and an optimal exercise time is given for the American put option. It is shown that the optimal fuzzy price is a solution of the optimality equation under a reasonable assumption. The permissible range of the writer’s (seller’s) optimal expected price in the American put option is presented, and the meaning and properties of the optimal expected prices are discussed in numerical examples. In a numerical example, the discrete-time approximation model is discussed for the continuous-time model.

MSC:
91G20 Derivative securities (option pricing, hedging, etc.)
60G40 Stopping times; optimal stopping problems; gambling theory
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[1] Bellman, R.E.; Zadeh, L.A., Decision-making in a fuzzy environment, Management science series B, 17, 141-164, (1970) · Zbl 0224.90032
[2] Blumenthal, R.M.; Getoor, R.K., Markov processes and potential theory, (1968), Academic Press New York · Zbl 0169.49204
[3] Cox, J.C.; Ross, S.A.; Rubinstein, M., Option pricing: A simplified approach, Journal of financial economics, 7, 229-263, (1979) · Zbl 1131.91333
[4] Elliott, R.J.; Kopp, P.E., Mathematics of financial markets, (1999), Springer New York · Zbl 0943.91035
[5] Klir, G.J.; Yuan, B., Fuzzy sets and fuzzy logic: theory and applications, (1995), Prentice-Hall London · Zbl 0915.03001
[6] Kurano, M.; Yasuda, M.; Nakagami, J.; Yoshida, Y., A limit theorem in some dynamic fuzzy systems, Fuzzy sets and systems, 51, 83-88, (1992) · Zbl 0796.93076
[7] Kurano, M.; Yasuda, M.; Nakagami, J.; Yoshida, Y., Markov-type fuzzy decision processes with a discounted reward on a closed interval, European journal of operational research, 92, 649-662, (1996) · Zbl 0914.90264
[8] Merton, R.C., Continuous-time finance, (1990), Blackwell Cambridge, MA
[9] Neveu, J., Discrete-parameter martingales, (1975), North-Holland New York · Zbl 0345.60026
[10] Puri, M.L.; Ralescu, D.A., Fuzzy random variables, Journal of mathematical analysis and applications, 114, 409-422, (1986) · Zbl 0592.60004
[11] Pliska, S.R., Introduction to mathematical finance: discrete-time models, (1997), Blackwell Publisher New York
[12] Ross, S.M., An introduction to mathematical finance, (1999), Cambridge University Press Cambridge · Zbl 0944.91024
[13] M. Sugeno, Theory of fuzzy integrals and its applications, Doctoral Thesis, Tokyo Institute of Technology, 1974
[14] Wang, G.; Zhang, Y., The theory of fuzzy stochastic processes, Fuzzy sets and systems, 51, 161-178, (1992) · Zbl 0782.60039
[15] Yoshida, Y., An optimal stopping problem in dynamic fuzzy systems with fuzzy rewards, Computers and mathematics with applications, 32, 17-28, (1996) · Zbl 0870.60040
[16] Yoshida, Y., Fuzzy decision processes with expected fuzzy rewards, (), 313-323, (Chapter 21) · Zbl 0893.90176
[17] Yoshida, Y., A time-average fuzzy reward criterion in fuzzy decision processes, Information sciences, 110, 103-112, (1998) · Zbl 0951.90056
[18] Yoshida, Y.; Yasuda, M.; Nakagami, J.; Kurano, M., Optimal stopping problems in a stochastic and fuzzy system, Journal of mathematical analysis and applications, 246, 135-149, (2000) · Zbl 0974.60024
[19] Zadeh, L.A., Fuzzy sets, Information and control, 8, 338-353, (1965) · Zbl 0139.24606
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